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a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).
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%I #9 Jan 28 2019 10:25:39

%S 1,3,7,16,40,119,450,2253,15207,139190,1731703,29335875,677864041,

%T 21400069232,924419728471,54716596051100,4443400439075834,

%U 495676372493566749,76041424515817042402,16060385520094706930608,4674665948889147697184915

%N a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).

%C Number of multiset partitions of integer partitions of 2^(n - 1) whose parts are constant and have equal sums.

%H Seiichi Manyama, <a href="/A323776/b323776.txt">Table of n, a(n) for n = 1..120</a>

%e The a(1) = 1 through a(4) = 16 partitions of partitions:

%e (1) (2) (4) (8)

%e (11) (22) (44)

%e (1)(1) (1111) (2222)

%e (2)(2) (4)(4)

%e (2)(11) (4)(22)

%e (11)(11) (22)(22)

%e (1)(1)(1)(1) (4)(1111)

%e (11111111)

%e (22)(1111)

%e (1111)(1111)

%e (2)(2)(2)(2)

%e (2)(2)(2)(11)

%e (2)(2)(11)(11)

%e (2)(11)(11)(11)

%e (11)(11)(11)(11)

%e (1)(1)(1)(1)(1)(1)(1)(1)

%t Table[Sum[Binomial[k+2^(n-k)-1,k-1],{k,n}],{n,20}]

%o (PARI) a(n) = sum(k=1, n, binomial(k+2^(n-k)-1, k-1)); \\ _Michel Marcus_, Jan 28 2019

%Y Cf. A000123, A001970, A002577, A006171, A007716, A034729, A047968, A279787, A279789, A305551, A306017, A319056, A323766, A323774, A323775.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 27 2019